Verifying probabilistic systems: new algorithms and complexity results

The content of the dissertation falls in the area of formal verification of probabilistic systems. It comprises four parts listed below: 1. the decision problem of (probabilistic) simulation preorder between probabilistic pushdown automata (pPDAs) and finite probabilistic automata (fPAs); 2. the decision problem of a bisimilarity metric on finite probabilistic automata (fPAs); 3. the approximation problem of acceptance probability of deterministictimed-automata (DTA) objectives on continuous-time Markov chains (CTMCs); 4. the approximation problem of cost-bounded reachability probability on continuous-time Markov decision processes (CTMDPs). The first two parts are concerned with equivalence checking on probabilistic automata, where probabilistic automata (PAs) are an analogue of discretetime Markov decision processes that involves both non-determinism and discrete-time stochastic transitions. The last two parts are concerned with numerical algorithms on Markov jump processes. In Part 1 and Part 2, we mainly focus on complexity issues; as for Part 3 and Part 4, we mainly focus on numerical approximation algorithms. In Part 1, we prove that the decision problem of (probabilistic) simulation preorder between pPDAs and fPAs is in EXPTIME. A pPDA is a pushdown automaton extended with probabilistic transitions, and generally it induces an infinite-state PA. The simulation preorder is a preorder that characterizes whether one probabilistic process (modelled as a PA) can mimic the other; technically speaking, it is the one-sided version of (probabilistic) bisimulation, which instead characterizes whether two probabilistic processes are behaviourally equivalent. We demonstrate the EXPTIME-membership of the decision problem through a tableaux system and a partition-refinement algorithm. Combined with the EXPTIMEhardness result by Kučera and Mayr (2010), we are able to show that the decision problem is EXPTIME-complete. The complexity result coincides with the one by Kučera and Mayr (2010) on non-probabilistic pushdown automata. Moreover, we obtain a fixed-parameter-tractable result on this